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<title>函数的单调性与最值</title>
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<body><h3 id='一函数单调性的判断与证明'>一、函数单调性的判断与证明</h3>
<h4 id='2-下列说法中正确的有'>2. 下列说法中，正确的有</h4>
<h4 id='①-若任意的\n x1x2∈i \n当\n x1x2 \n时\n fx1−fx2x1−x20 \n则\n y−fx \n在\n i \n上是增函数'>① 若任意的<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.607ex" height="1.984ex" role="img" focusable="false" viewBox="0 -683 4688.3 877" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.439ex;"><defs><path id="MJX-71-TEX-N-A0" d=""></path><path id="MJX-71-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 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38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="A0" xlink:href="#MJX-71-TEX-N-A0"></use></g><g data-mml-node="msub" transform="translate(250,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-71-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-71-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1258.6,0)"><use data-c="2C" xlink:href="#MJX-71-TEX-N-2C"></use></g><g data-mml-node="msub" transform="translate(1703.2,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-71-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-71-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(2989.6,0)"><use data-c="2208" xlink:href="#MJX-71-TEX-N-2208"></use></g><g data-mml-node="mi" transform="translate(3934.3,0)"><use data-c="1D43C" xlink:href="#MJX-71-TEX-I-1D43C"></use></g><g data-mml-node="mtext" transform="translate(4438.3,0)"><use data-c="A0" xlink:href="#MJX-71-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ x_1,x_2\in I\ </script>当<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.712ex" height="1.561ex" role="img" focusable="false" viewBox="0 -540 3850.7 690" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.339ex;"><defs><path id="MJX-72-TEX-N-A0" d=""></path><path id="MJX-72-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-72-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-72-TEX-N-3C" d="M694 -11T694 -19T688 -33T678 -40Q671 -40 524 29T234 166L90 235Q83 240 83 250Q83 261 91 266Q664 540 678 540Q681 540 687 534T694 519T687 505Q686 504 417 376L151 250L417 124Q686 -4 687 -5Q694 -11 694 -19Z"></path><path id="MJX-72-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 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data-mml-node="msub" transform="translate(2592.1,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-72-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-72-TEX-N-32"></use></g></g><g data-mml-node="mtext" transform="translate(3600.7,0)"><use data-c="A0" xlink:href="#MJX-72-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ x_1<x_2\ </script>时，<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="14.996ex" height="3.39ex" role="img" focusable="false" viewBox="0 -1047.1 6628.1 1498.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -1.021ex;"><defs><path id="MJX-73-TEX-N-A0" d=""></path><path id="MJX-73-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-73-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-73-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path><path id="MJX-73-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-73-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 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xlink:href="#MJX-73-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ \frac{f(x_1)-f(x_2)}{x_1-x_2}>0\ </script>，则<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.304ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 4112.4 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-74-TEX-N-A0" d=""></path><path id="MJX-74-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 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681 359 681Q481 681 487 683H497Q504 676 504 672T501 655T494 639Q491 637 471 637Q440 637 407 634Q393 631 388 623Q381 609 337 432Q326 385 315 341Q245 65 245 59Q245 52 255 50T307 46H339Q345 38 345 37T342 19Q338 6 332 0H316Q279 2 179 2Q143 2 113 2T65 2T43 1Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="A0" xlink:href="#MJX-75-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(250,0)"><use data-c="1D43C" xlink:href="#MJX-75-TEX-I-1D43C"></use></g><g data-mml-node="mtext" transform="translate(754,0)"><use data-c="A0" xlink:href="#MJX-75-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ I\ </script>上是增函数；</h4>
<h4 id='②-函数\n yx2 \n在\n r \n上是增函数'>② 函数<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.539ex" height="2.351ex" role="img" focusable="false" viewBox="0 -833.9 3332.1 1038.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.464ex;"><defs><path id="MJX-76-TEX-N-A0" d=""></path><path id="MJX-76-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 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transform="translate(2073.6,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-76-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,363) scale(0.707)"><use data-c="32" xlink:href="#MJX-76-TEX-N-32"></use></g></g><g data-mml-node="mtext" transform="translate(3082.1,0)"><use data-c="A0" xlink:href="#MJX-76-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ y=x^2\ </script>在<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.848ex" height="1.593ex" role="img" focusable="false" viewBox="0 -683 1259 704" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.048ex;"><defs><path id="MJX-98-TEX-N-A0" d=""></path><path id="MJX-98-TEX-I-1D445" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="A0" xlink:href="#MJX-98-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(250,0)"><use data-c="1D445" xlink:href="#MJX-98-TEX-I-1D445"></use></g><g data-mml-node="mtext" transform="translate(1009,0)"><use data-c="A0" xlink:href="#MJX-98-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ R\ </script>上是增函数；</h4>
<h4 id='③-函数\n y−1x \n在定义域上是增函数'>③ 函数<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="8.928ex" height="2.755ex" role="img" focusable="false" viewBox="0 -864.9 3946 1217.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.798ex;"><defs><path id="MJX-78-TEX-N-A0" d=""></path><path id="MJX-78-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 82Q103 89 103 113Q103 170 138 262T173 379Q173 380 173 381Q173 390 173 393T169 400T158 404H154Q131 404 112 385T82 344T65 302T57 280Q55 278 41 278H27Q21 284 21 287Z"></path><path id="MJX-78-TEX-N-3D" d="M56 347Q56 360 70 367H707Q722 359 722 347Q722 336 708 328L390 327H72Q56 332 56 347ZM56 153Q56 168 72 173H708Q722 163 722 153Q722 140 707 133H70Q56 140 56 153Z"></path><path id="MJX-78-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-78-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-78-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 153H489Q504 153 504 145Q504 144 502 134Q486 77 440 33T333 -11Q263 -11 227 52Q186 -10 133 -10H127Q78 -10 57 16T35 71Q35 103 54 123T99 143Q142 143 142 101Q142 81 130 66T107 46T94 41L91 40Q91 39 97 36T113 29T132 26Q168 26 194 71Q203 87 217 139T245 247T261 313Q266 340 266 352Q266 380 251 392T217 404Q177 404 142 372T93 290Q91 281 88 280T72 278H58Q52 284 52 289Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="A0" xlink:href="#MJX-78-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(250,0)"><use data-c="1D466" xlink:href="#MJX-78-TEX-I-1D466"></use></g><g data-mml-node="mo" transform="translate(1017.8,0)"><use data-c="3D" xlink:href="#MJX-78-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(2073.6,0)"><use data-c="2212" xlink:href="#MJX-78-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(2851.6,0)"><g data-mml-node="mn" transform="translate(245.5,394) scale(0.707)"><use data-c="31" xlink:href="#MJX-78-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(220,-345) scale(0.707)"><use data-c="1D465" xlink:href="#MJX-78-TEX-I-1D465"></use></g><rect width="604.5" height="60" x="120" y="220"></rect></g><g data-mml-node="mtext" transform="translate(3696,0)"><use data-c="A0" xlink:href="#MJX-78-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ y=-\frac1x\ </script>在定义域上是增函数;</h4>
<h4 id='④-函数\n y1x \n的单调区间是\n\n\n −∞0∪0∞'>④ 函数<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.167ex" height="2.755ex" role="img" focusable="false" viewBox="0 -864.9 3168 1217.7" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.798ex;"><defs><path id="MJX-79-TEX-N-A0" d=""></path><path id="MJX-79-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 231 -11Q185 -11 150 11T104 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<ul>
<li><input type='checkbox' /><h5 id='a-0-个'>A. 0 个</h5>
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<li><input type='checkbox' /><h5 id='b-1-个'>B. 1 个</h5>
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<li><input type='checkbox' checked/><h5 id='c-2-个'>C. 2 个</h5>
</li>
<li><input type='checkbox' /><h5 id='d-3-个'>D. 3 个</h5>
</li>

</ul>
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transform="translate(16479.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g></g></g></g></g></g></g></g></g></g></svg></mjx-container><script type="math/tex">\color{blue}\begin{align}&设\ x_1，x_2\ 是区间\ (-1，+\infty)\ 上任意两个实数且\ x_1<x_2则\\&f(x_1)-f(x_2)=\frac{x_1+2}{x_1+1}-\frac{x_2+2}{x_2+1}=\frac{x_2-x_1}{(x_1+1)(x_2+1)}\\&\because -1<x_1<x_2\\&\therefore x_2-x_1>0,x_1+1>0,x_2+1>0\\&\therefore\frac{x_2-x_1}{(x_1+1)(x_2+1)}>0\\&\ 即\ f(x_1)-f(x_2)>0,f(x_1)>f(x_2)\\&\therefore y=\frac{x+2}{x+1}\ 在（-1,\ +\infty)上是减函数\end{align}</script></p>
<p>&nbsp;</p>
<h3 id='二函数的单调区间'>二、函数的单调区间</h3>
<h4 id='5-求下列函数的单调区间'>5. 求下列函数的单调区间</h4>
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<p><mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="27.603ex" height="8.369ex" role="img" focusable="false" viewBox="0 -2099.5 12200.7 3699" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -3.619ex;"><defs><path id="MJX-85-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 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xlink:href="#MJX-85-TEX-I-1D465"></use></g></g><g data-mml-node="mtd" transform="translate(2850,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-85-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="3C" xlink:href="#MJX-85-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(1905.6,0)"><use data-c="30" xlink:href="#MJX-85-TEX-N-30"></use></g></g></g></g><g data-mml-node="mo" transform="translate(6005.6,0) translate(0 250)"></g></g></g></g><g data-mml-node="mtr" transform="translate(0,-1349.5)"><g data-mml-node="mtd"></g><g data-mml-node="mtd"><g data-mml-node="mtext"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">图</text></g><g data-mml-node="mtext" transform="translate(884,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">象</text></g><g data-mml-node="mtext" transform="translate(1768,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">如</text></g><g data-mml-node="mtext" transform="translate(2652,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">图</text></g><g data-mml-node="mtext" transform="translate(3536,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">所</text></g><g data-mml-node="mtext" transform="translate(4420,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">示</text></g></g></g></g></g></g></g></g></g></g></svg></mjx-container><script type="math/tex">\color{blue}\begin{align}&f(x)=3|x|=\begin{cases}3x&x\geqslant0\\-3x&x<0\end{cases}\\&图象如图所示\end{align}</script></p>
<p style="width:250px"><img src="二-5-1.png"></p>
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font-family="serif">单</text></g><g data-mml-node="mtext" transform="translate(3918,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">调</text></g><g data-mml-node="mtext" transform="translate(4802,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">递</text></g><g data-mml-node="mtext" transform="translate(5686,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">减</text></g><g data-mml-node="mtext" transform="translate(6570,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">区</text></g><g data-mml-node="mtext" transform="translate(7454,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">间</text></g><g data-mml-node="mtext" transform="translate(8338,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">为</text></g><g data-mml-node="mo" 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transform="translate(18860.3,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">为</text></g><g data-mml-node="mo" transform="translate(19744.3,0)"><use data-c="5B" xlink:href="#MJX-86-TEX-N-5B"></use></g><g data-mml-node="mn" transform="translate(20022.3,0)"><use data-c="30" xlink:href="#MJX-86-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(20522.3,0)"><use data-c="2C" xlink:href="#MJX-86-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(20967,0)"><use data-c="A0" xlink:href="#MJX-86-TEX-N-A0"></use></g><g data-mml-node="mo" transform="translate(21439.2,0)"><use data-c="2B" xlink:href="#MJX-86-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(22439.4,0)"><use data-c="221E" xlink:href="#MJX-86-TEX-N-221E"></use></g><g data-mml-node="mo" transform="translate(23439.4,0)"><use data-c="29" xlink:href="#MJX-86-TEX-N-29"></use></g></g></g></g></svg></mjx-container><script type="math/tex">\color{blue}f(x)\ 的单调递减区间为(-\infty,\ 0],\ 单调递增区间为[0,\ +\infty)</script></p>
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transform="translate(13337,0)"><g data-mml-node="mo"><use data-c="29" xlink:href="#MJX-88-TEX-N-29"></use></g><g data-mml-node="mn" transform="translate(422,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-88-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(14384.8,0)"><use data-c="2212" xlink:href="#MJX-88-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(15385,0)"><use data-c="34" xlink:href="#MJX-88-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(15885,0)"><use data-c="2C" xlink:href="#MJX-88-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(16329.7,0)"><use data-c="A0" xlink:href="#MJX-88-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(16579.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">则</text></g><g data-mml-node="mtext" transform="translate(17463.7,0)"><use data-c="A0" xlink:href="#MJX-88-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(17713.7,0)"><use data-c="1D453" xlink:href="#MJX-88-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(18263.7,0)"><use data-c="28" xlink:href="#MJX-88-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(18652.7,0)"><use data-c="1D465" xlink:href="#MJX-88-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(19224.7,0)"><use data-c="29" xlink:href="#MJX-88-TEX-N-29"></use></g><g data-mml-node="mtext" transform="translate(19613.7,0)"><use data-c="A0" xlink:href="#MJX-88-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(19863.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">的</text></g><g data-mml-node="mtext" transform="translate(20747.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">图</text></g><g data-mml-node="mtext" transform="translate(21631.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">象</text></g><g data-mml-node="mtext" transform="translate(22515.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">如</text></g><g data-mml-node="mtext" transform="translate(23399.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">图</text></g><g data-mml-node="mtext" transform="translate(24283.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">所</text></g><g data-mml-node="mtext" transform="translate(25167.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">示</text></g></g></g></g></g></g></g></svg></mjx-container><script type="math/tex">\color{blue}\begin{align}令\ g(x)=x^2+2x-3=(x+1)^2-4,\ 则\ f(x)\ 的图象如图所示\end{align}</script></p>
<p style="width:250px"><img src="二-5-2.png"></p>
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font-family="serif">易</text></g><g data-mml-node="mtext" transform="translate(884,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">得</text></g><g data-mml-node="mo" transform="translate(1768,0)"><use data-c="2C" xlink:href="#MJX-89-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(2212.7,0)"><use data-c="A0" xlink:href="#MJX-89-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(2462.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">函</text></g><g data-mml-node="mtext" transform="translate(3346.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g><g data-mml-node="mtext" transform="translate(4230.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">的</text></g><g data-mml-node="mtext" transform="translate(5114.7,0)"><text data-variant="normal" transform="scale(1,-1)" 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<h4 id='7-设函数\n yxaxbab \n求\n fx \n的单调区间并说明\n fx \n在其单调区间上的单调性'>7. 设函数<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="16.203ex" height="2.745ex" role="img" focusable="false" viewBox="0 -810.2 7161.8 1213.2" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.912ex;"><defs><path id="MJX-90-TEX-N-A0" d=""></path><path id="MJX-90-TEX-I-1D466" d="M21 287Q21 301 36 335T84 406T158 442Q199 442 224 419T250 355Q248 336 247 334Q247 331 231 288T198 191T182 105Q182 62 196 45T238 27Q261 27 281 38T312 61T339 94Q339 95 344 114T358 173T377 247Q415 397 419 404Q432 431 462 431Q475 431 483 424T494 412T496 403Q496 390 447 193T391 -23Q363 -106 294 -155T156 -205Q111 -205 77 -183T43 -117Q43 -95 50 -80T69 -58T89 -48T106 -45Q150 -45 150 -87Q150 -107 138 -122T115 -142T102 -147L99 -148Q101 -153 118 -160T152 -167H160Q177 -167 186 -165Q219 -156 247 -127T290 -65T313 -9T321 21L315 17Q309 13 296 6T270 -6Q250 -11 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578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 -132Q55 -95 76 -78T118 -61Q162 -61 162 -103Q162 -122 151 -136T127 -157L118 -162Z"></path><path id="MJX-92-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-92-TEX-I-1D465" d="M52 289Q59 331 106 386T222 442Q257 442 286 424T329 379Q371 442 430 442Q467 442 494 420T522 361Q522 332 508 314T481 292T458 288Q439 288 427 299T415 328Q415 374 465 391Q454 404 425 404Q412 404 406 402Q368 386 350 336Q290 115 290 78Q290 50 306 38T341 26Q378 26 414 59T463 140Q466 150 469 151T485 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font-size="884px" font-family="serif">均</text></g><g data-mml-node="mtext" transform="translate(21471.2,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">为</text></g><g data-mml-node="mtext" transform="translate(22355.2,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">减</text></g><g data-mml-node="mtext" transform="translate(23239.2,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">函</text></g><g data-mml-node="mtext" transform="translate(24123.2,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g></g></g></g></g></g></g></g></g></g></svg></mjx-container><script type="math/tex">\color{blue}\begin{align}&函数\ f(x)\ 的定义域为\ (-\infty,\ -b)\cup(-b,\ +\infty)\\&任取\ x_1,\ x_2\in(-\infty,\ -b).\ 且\ x_1<x_2\\&则\ f(x_2)-f(x_1)=\frac{x_2+a}{x_2+b}-\frac{x_1+a}{x_1+b}\\&\qquad\qquad\quad\quad\ \ \ \ =\frac{(x_2-x_1)(b-a)}{(x_2+b)(x_1+b)}\\&\because  x_1<x_2<-b,\ a>b\\&\therefore x_2-x_1>0,\ b-a<0\\&\quad\ x_2+b<0,\ x_1+b<0\\&\therefore\frac{(x_2-x_1)(b-a)}{(x_2+b)(x_1+b)}<0,\ 即f(x_2)-f(x_1)<0\\&\ 即f(x_2)<f(x_1)\\&\therefore f(x)在\ (-\infty,\ -b)\ 上单调递减\\&同理可证\ f(x)\ 在\ (-b,\ +\infty)\ 上单调递减\\&故\ f(x)=\frac{x+a}{x+b}在\ (-\infty,\ -b)\ 和\ (-b,\ +\infty)\ 上均为减函数\end{align}</script></p>
<p>&nbsp;</p>
<h3 id='三复合函数的单调性'>三、复合函数的单调性</h3>
<h4 id='8-已知函数\n\n fx82x−x2gxf2−x2 \n试求\n gx \n的单调区间'>8. 已知函数<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="38.151ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 16862.6 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-94-TEX-N-A0" d=""></path><path id="MJX-94-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 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xlink:href="#MJX-94-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ f(x)=8+2x-x^2，g(x)=f(2-x^2)\ </script>，试求<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="5.265ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2327 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-95-TEX-N-A0" d=""></path><path id="MJX-95-TEX-I-1D454" d="M311 43Q296 30 267 15T206 0Q143 0 105 45T66 160Q66 265 143 353T314 442Q361 442 401 394L404 398Q406 401 409 404T418 412T431 419T447 422Q461 422 470 413T480 394Q480 379 423 152T363 -80Q345 -134 286 -169T151 -205Q10 -205 10 -137Q10 -111 28 -91T74 -71Q89 -71 102 -80T116 -111Q116 -121 114 -130T107 -144T99 -154T92 -162L90 -164H91Q101 -167 151 -167Q189 -167 211 -155Q234 -144 254 -122T282 -75Q288 -56 298 -13Q311 35 311 43ZM384 328L380 339Q377 350 375 354T369 368T359 382T346 393T328 402T306 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font-family="serif">令</text></g><g data-mml-node="mtext" transform="translate(884,0)"><use data-c="A0" xlink:href="#MJX-96-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(1134,0)"><use data-c="1D462" xlink:href="#MJX-96-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(1706,0)"><use data-c="28" xlink:href="#MJX-96-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(2095,0)"><use data-c="1D465" xlink:href="#MJX-96-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(2667,0)"><use data-c="29" xlink:href="#MJX-96-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(3333.8,0)"><use data-c="3D" xlink:href="#MJX-96-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(4389.6,0)"><use data-c="32" xlink:href="#MJX-96-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(5111.8,0)"><use data-c="2212" xlink:href="#MJX-96-TEX-N-2212"></use></g><g data-mml-node="msup" transform="translate(6112,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-96-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,413) scale(0.707)"><use data-c="32" xlink:href="#MJX-96-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(7120.6,0)"><use data-c="2C" xlink:href="#MJX-96-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(7565.2,0)"><use data-c="A0" xlink:href="#MJX-96-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(7815.2,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">则</text></g><g data-mml-node="mtext" transform="translate(8699.2,0)"><use data-c="A0" xlink:href="#MJX-96-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(8949.2,0)"><use data-c="1D462" xlink:href="#MJX-96-TEX-I-1D462"></use></g><g data-mml-node="mo" transform="translate(9521.2,0)"><use data-c="28" xlink:href="#MJX-96-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(9910.2,0)"><use 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data-c="221E" xlink:href="#MJX-96-TEX-N-221E"></use></g><g data-mml-node="mo" transform="translate(13828,0)"><use data-c="2C" xlink:href="#MJX-96-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(14272.7,0)"><use data-c="A0" xlink:href="#MJX-96-TEX-N-A0"></use></g><g data-mml-node="mn" transform="translate(14522.7,0)"><use data-c="31" xlink:href="#MJX-96-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(15022.7,0)"><use data-c="5D" xlink:href="#MJX-96-TEX-N-5D"></use></g><g data-mml-node="mtext" transform="translate(15300.7,0)"><use data-c="A0" xlink:href="#MJX-96-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(15550.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">上</text></g><g data-mml-node="mtext" transform="translate(16434.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">为</text></g><g data-mml-node="mtext" transform="translate(17318.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">增</text></g><g data-mml-node="mtext" transform="translate(18202.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">函</text></g><g data-mml-node="mtext" transform="translate(19086.7,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">数</text></g><g data-mml-node="mo" transform="translate(19970.7,0)"><use data-c="2C" xlink:href="#MJX-96-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(20415.3,0)"><use data-c="A0" xlink:href="#MJX-96-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(20665.3,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">在</text></g><g data-mml-node="mtext" transform="translate(21549.3,0)"><use data-c="A0" xlink:href="#MJX-96-TEX-N-A0"></use></g><g data-mml-node="mo" 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data-mml-node="mi" transform="translate(7365.7,0)"><use data-c="1D465" xlink:href="#MJX-96-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(8215.4,0)"><use data-c="3D" xlink:href="#MJX-96-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(9271.2,0)"><use data-c="B1" xlink:href="#MJX-96-TEX-N-B1"></use></g><g data-mml-node="mn" transform="translate(10049.2,0)"><use data-c="31" xlink:href="#MJX-96-TEX-N-31"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-2150.9)"><g data-mml-node="mtd"></g><g data-mml-node="mtd"></g></g></g></g></g></g></g></g></g></svg></mjx-container><script type="math/tex">\color{blue}\begin{align}&令\ u(x)=2-x^2,\ 则\ u(x)\ 在\ (-\infty,\ 0]\ 上为增函数,\ 在[0,\ +\infty)\ 上为减函数\\&又知\ f(x)=8+2x-x^2\ 在\ (-\infty,\ 1]\ 上为增函数,\ 在\ [1,\ +\infty)\ 上为减函数\\&令-x^2+2=1,\ x=\pm1\\&\end{align}</script></p>
<p>&nbsp;</p>
<h3 id='四函数单调性的应用'>四、函数单调性的应用</h3>
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xlink:href="#MJX-97-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(5823.9,0)"><use data-c="1D44E" xlink:href="#MJX-97-TEX-I-1D44E"></use></g></g><g data-mml-node="mtd" transform="translate(7352.9,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-97-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="3C" xlink:href="#MJX-97-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(1905.6,0)"><use data-c="31" xlink:href="#MJX-97-TEX-N-31"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-600)"><g data-mml-node="mtd"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-97-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(778,0)"><use data-c="1D465" xlink:href="#MJX-97-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(1572.2,0)"><use data-c="2B" xlink:href="#MJX-97-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2572.4,0)"><use data-c="31" xlink:href="#MJX-97-TEX-N-31"></use></g></g><g data-mml-node="mtd" transform="translate(7352.9,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-97-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(849.8,0)"><use data-c="2A7E" xlink:href="#MJX-97-TEX-N-2A7E"></use></g><g data-mml-node="mn" transform="translate(1905.6,0)"><use data-c="31" xlink:href="#MJX-97-TEX-N-31"></use></g></g></g></g><g data-mml-node="mo" transform="translate(10508.4,0) translate(0 250)"></g></g><g data-mml-node="mtext" transform="translate(12825.1,0)"><use data-c="A0" xlink:href="#MJX-97-TEX-N-A0"></use></g></g></g></g></g></g></svg></mjx-container><script type="math/tex">\ f(x)\begin{cases}(3a-1)x+4a&x<1\\-x+1&x\geqslant1\end{cases}\ </script>是定义在<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.848ex" height="1.593ex" role="img" focusable="false" viewBox="0 -683 1259 704" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.048ex;"><defs><path id="MJX-98-TEX-N-A0" d=""></path><path id="MJX-98-TEX-I-1D445" d="M230 637Q203 637 198 638T193 649Q193 676 204 682Q206 683 378 683Q550 682 564 680Q620 672 658 652T712 606T733 563T739 529Q739 484 710 445T643 385T576 351T538 338L545 333Q612 295 612 223Q612 212 607 162T602 80V71Q602 53 603 43T614 25T640 16Q668 16 686 38T712 85Q717 99 720 102T735 105Q755 105 755 93Q755 75 731 36Q693 -21 641 -21H632Q571 -21 531 4T487 82Q487 109 502 166T517 239Q517 290 474 313Q459 320 449 321T378 323H309L277 193Q244 61 244 59Q244 55 245 54T252 50T269 48T302 46H333Q339 38 339 37T336 19Q332 6 326 0H311Q275 2 180 2Q146 2 117 2T71 2T50 1Q33 1 33 10Q33 12 36 24Q41 43 46 45Q50 46 61 46H67Q94 46 127 49Q141 52 146 61Q149 65 218 339T287 628Q287 635 230 637ZM630 554Q630 586 609 608T523 636Q521 636 500 636T462 637H440Q393 637 386 627Q385 624 352 494T319 361Q319 360 388 360Q466 361 492 367Q556 377 592 426Q608 449 619 486T630 554Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="A0" xlink:href="#MJX-98-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(250,0)"><use data-c="1D445" xlink:href="#MJX-98-TEX-I-1D445"></use></g><g data-mml-node="mtext" transform="translate(1009,0)"><use data-c="A0" xlink:href="#MJX-98-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ R\ </script>上的减函数，那么<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="2.328ex" height="1.02ex" role="img" focusable="false" viewBox="0 -441 1029 451" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.023ex;"><defs><path id="MJX-106-TEX-N-A0" d=""></path><path id="MJX-106-TEX-I-1D44E" d="M33 157Q33 258 109 349T280 441Q331 441 370 392Q386 422 416 422Q429 422 439 414T449 394Q449 381 412 234T374 68Q374 43 381 35T402 26Q411 27 422 35Q443 55 463 131Q469 151 473 152Q475 153 483 153H487Q506 153 506 144Q506 138 501 117T481 63T449 13Q436 0 417 -8Q409 -10 393 -10Q359 -10 336 5T306 36L300 51Q299 52 296 50Q294 48 292 46Q233 -10 172 -10Q117 -10 75 30T33 157ZM351 328Q351 334 346 350T323 385T277 405Q242 405 210 374T160 293Q131 214 119 129Q119 126 119 118T118 106Q118 61 136 44T179 26Q217 26 254 59T298 110Q300 114 325 217T351 328Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="A0" xlink:href="#MJX-106-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(250,0)"><use data-c="1D44E" xlink:href="#MJX-106-TEX-I-1D44E"></use></g><g data-mml-node="mtext" transform="translate(779,0)"><use data-c="A0" xlink:href="#MJX-106-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ a\ </script>的取值范围是</h4>
<ul>
<li><input type='checkbox' /><h4 id='a-\n\n−∞13'>A. <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="9.578ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 4233.6 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.816ex;"><defs><path id="MJX-100-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-100-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-100-TEX-N-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z"></path><path id="MJX-100-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-100-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path id="MJX-100-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-100-TEX-N-28"></use></g><g data-mml-node="mo" transform="translate(389,0)"><use data-c="2212" xlink:href="#MJX-100-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(1167,0)"><use data-c="221E" xlink:href="#MJX-100-TEX-N-221E"></use></g><g data-mml-node="mi" transform="translate(2167,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g><g data-mml-node="mfrac" transform="translate(3051,0)"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJX-100-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="33" xlink:href="#MJX-100-TEX-N-33"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3844.6,0)"><use data-c="29" xlink:href="#MJX-100-TEX-N-29"></use></g></g></g></svg></mjx-container><script type="math/tex">(-\infty，\frac13)</script></h4>
</li>
<li><input type='checkbox' /><h5 id='b-\n\n17∞'>B. <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="10.584ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 4678 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.816ex;"><defs><path id="MJX-101-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-101-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-101-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path id="MJX-101-TEX-N-2B" d="M56 237T56 250T70 270H369V420L370 570Q380 583 389 583Q402 583 409 568V270H707Q722 262 722 250T707 230H409V-68Q401 -82 391 -82H389H387Q375 -82 369 -68V230H70Q56 237 56 250Z"></path><path id="MJX-101-TEX-N-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z"></path><path id="MJX-101-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="28" xlink:href="#MJX-101-TEX-N-28"></use></g><g data-mml-node="mfrac" transform="translate(389,0)"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJX-101-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="37" xlink:href="#MJX-101-TEX-N-37"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mi" transform="translate(1182.6,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g><g data-mml-node="mo" transform="translate(2288.8,0)"><use data-c="2B" xlink:href="#MJX-101-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(3289,0)"><use data-c="221E" xlink:href="#MJX-101-TEX-N-221E"></use></g><g data-mml-node="mo" transform="translate(4289,0)"><use data-c="29" xlink:href="#MJX-101-TEX-N-29"></use></g></g></g></svg></mjx-container><script type="math/tex">(\frac17，+\infty)</script></h5>
</li>
<li><input type='checkbox' checked/><h5 id='c-\n\n1713'>C. <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="7.1ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 3138.1 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.816ex;"><defs><path id="MJX-102-TEX-N-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path id="MJX-102-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-102-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 458V455H55V458Z"></path><path id="MJX-102-TEX-N-33" d="M127 463Q100 463 85 480T69 524Q69 579 117 622T233 665Q268 665 277 664Q351 652 390 611T430 522Q430 470 396 421T302 350L299 348Q299 347 308 345T337 336T375 315Q457 262 457 175Q457 96 395 37T238 -22Q158 -22 100 21T42 130Q42 158 60 175T105 193Q133 193 151 175T169 130Q169 119 166 110T159 94T148 82T136 74T126 70T118 67L114 66Q165 21 238 21Q293 21 321 74Q338 107 338 175V195Q338 290 274 322Q259 328 213 329L171 330L168 332Q166 335 166 348Q166 366 174 366Q202 366 232 371Q266 376 294 413T322 525V533Q322 590 287 612Q265 626 240 626Q208 626 181 615T143 592T132 580H135Q138 579 143 578T153 573T165 566T175 555T183 540T186 520Q186 498 172 481T127 463Z"></path><path id="MJX-102-TEX-N-29" d="M60 749L64 750Q69 750 74 750H86L114 726Q208 641 251 514T294 250Q294 182 284 119T261 12T224 -76T186 -143T145 -194T113 -227T90 -246Q87 -249 86 -250H74Q66 -250 63 -250T58 -247T55 -238Q56 -237 66 -225Q221 -64 221 250T66 725Q56 737 55 738Q55 746 60 749Z"></path></defs><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo"><use data-c="5B" xlink:href="#MJX-102-TEX-N-5B"></use></g><g data-mml-node="mfrac" transform="translate(278,0)"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJX-102-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="37" xlink:href="#MJX-102-TEX-N-37"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mi" transform="translate(1071.6,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">，</text></g><g data-mml-node="mfrac" transform="translate(1955.6,0)"><g data-mml-node="mn" transform="translate(220,394) scale(0.707)"><use data-c="31" xlink:href="#MJX-102-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-345) scale(0.707)"><use data-c="33" xlink:href="#MJX-102-TEX-N-33"></use></g><rect width="553.6" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(2749.1,0)"><use data-c="29" xlink:href="#MJX-102-TEX-N-29"></use></g></g></g></svg></mjx-container><script type="math/tex">[\frac17，\frac13)</script></h5>
</li>
<li><input type='checkbox' /><h5 id='d-\n\n\n−∞−17∪13∞'>D. <mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="25.191ex" height="2.773ex" role="img" focusable="false" viewBox="0 -864.9 11134.4 1225.5" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.816ex;"><defs><path id="MJX-103-TEX-N-28" d="M94 250Q94 319 104 381T127 488T164 576T202 643T244 695T277 729T302 750H315H319Q333 750 333 741Q333 738 316 720T275 667T226 581T184 443T167 250T184 58T225 -81T274 -167T316 -220T333 -241Q333 -250 318 -250H315H302L274 -226Q180 -141 137 -14T94 250Z"></path><path id="MJX-103-TEX-N-2212" d="M84 237T84 250T98 270H679Q694 262 694 250T679 230H98Q84 237 84 250Z"></path><path id="MJX-103-TEX-N-221E" d="M55 217Q55 305 111 373T254 442Q342 442 419 381Q457 350 493 303L507 284L514 294Q618 442 747 442Q833 442 888 374T944 214Q944 128 889 59T743 -11Q657 -11 580 50Q542 81 506 128L492 147L485 137Q381 -11 252 -11Q166 -11 111 57T55 217ZM907 217Q907 285 869 341T761 397Q740 397 720 392T682 378T648 359T619 335T594 310T574 285T559 263T548 246L543 238L574 198Q605 158 622 138T664 94T714 61T765 51Q827 51 867 100T907 217ZM92 214Q92 145 131 89T239 33Q357 33 456 193L425 233Q364 312 334 337Q285 380 233 380Q171 380 132 331T92 214Z"></path><path id="MJX-103-TEX-N-31" d="M213 578L200 573Q186 568 160 563T102 556H83V602H102Q149 604 189 617T245 641T273 663Q275 666 285 666Q294 666 302 660V361L303 61Q310 54 315 52T339 48T401 46H427V0H416Q395 3 257 3Q121 3 100 0H88V46H114Q136 46 152 46T177 47T193 50T201 52T207 57T213 61V578Z"></path><path id="MJX-103-TEX-N-37" d="M55 458Q56 460 72 567L88 674Q88 676 108 676H128V672Q128 662 143 655T195 646T364 644H485V605L417 512Q408 500 387 472T360 435T339 403T319 367T305 330T292 284T284 230T278 162T275 80Q275 66 275 52T274 28V19Q270 2 255 -10T221 -22Q210 -22 200 -19T179 0T168 40Q168 198 265 368Q285 400 349 489L395 552H302Q128 552 119 546Q113 543 108 522T98 479L95 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type="math/tex">(-\infty，-\frac17]\cup(\frac13，+\infty)</script></h5>
</li>

</ul>
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scale(0.707)"><use data-c="32" xlink:href="#MJX-107-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(7909.6,0)"><use data-c="2208" xlink:href="#MJX-107-TEX-N-2208"></use></g><g data-mml-node="mi" transform="translate(8854.3,0)"><use data-c="1D445" xlink:href="#MJX-107-TEX-I-1D445"></use></g><g data-mml-node="mo" transform="translate(9613.3,0)"><use data-c="2C" xlink:href="#MJX-107-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(10058,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(10308,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">都</text></g><g data-mml-node="mtext" transform="translate(11192,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">有</text></g><g data-mml-node="mtext" transform="translate(12076,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mo" 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transform="translate(21119.5,0)"><use data-c="28" xlink:href="#MJX-107-TEX-N-28"></use></g><g data-mml-node="msub" transform="translate(21508.5,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-107-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-107-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(22517.1,0)"><use data-c="29" xlink:href="#MJX-107-TEX-N-29"></use></g><g data-mml-node="mo" transform="translate(22906.1,0)"><use data-c="5D" xlink:href="#MJX-107-TEX-N-5D"></use></g><g data-mml-node="mo" transform="translate(23461.9,0)"><use data-c="3C" xlink:href="#MJX-107-TEX-N-3C"></use></g><g data-mml-node="mn" transform="translate(24517.7,0)"><use data-c="30" xlink:href="#MJX-107-TEX-N-30"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,3044)"><g data-mml-node="mtd"></g><g data-mml-node="mtd"><g data-mml-node="mtext"><text data-variant="normal" 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data-mml-node="mi" transform="translate(4975,0)"><use data-c="1D465" xlink:href="#MJX-107-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(5547,0)"><use data-c="29" xlink:href="#MJX-107-TEX-N-29"></use></g><g data-mml-node="mtext" transform="translate(5936,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">在</text></g><g data-mml-node="mtext" transform="translate(6820,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(7070,0)"><use data-c="1D445" xlink:href="#MJX-107-TEX-I-1D445"></use></g><g data-mml-node="mtext" transform="translate(7829,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(8079,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">上</text></g><g data-mml-node="mtext" transform="translate(8963,0)"><text data-variant="normal" transform="scale(1,-1)" 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transform="translate(24942.4,0)"><use data-c="2C" xlink:href="#MJX-107-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(25387.1,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(25637.1,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">且</text></g><g data-mml-node="mtext" transform="translate(26521.1,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">有</text></g><g data-mml-node="mtext" transform="translate(27405.1,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(27655.1,0)"><use data-c="1D453" xlink:href="#MJX-107-TEX-I-1D453"></use></g><g data-mml-node="mi" transform="translate(28205.1,0)"><text data-variant="italic" transform="scale(1,-1)" font-size="884px" font-family="serif" font-style="italic">（</text></g><g data-mml-node="mi" transform="translate(29205.6,0)"><use 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data-mml-node="mtr" transform="translate(0,-1581)"><g data-mml-node="mtd"></g><g data-mml-node="mtd"><g data-mml-node="mstyle"><g data-mml-node="mspace"></g></g><g data-mml-node="mtext" transform="translate(1000,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(1250,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">要</text></g><g data-mml-node="mtext" transform="translate(2134,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">使</text></g><g data-mml-node="mtext" transform="translate(3018,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(3268,0)"><use data-c="1D453" xlink:href="#MJX-107-TEX-I-1D453"></use></g><g data-mml-node="mo" transform="translate(3818,0)"><use data-c="28" xlink:href="#MJX-107-TEX-N-28"></use></g><g data-mml-node="mi" transform="translate(4207,0)"><use data-c="1D465" xlink:href="#MJX-107-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(4779,0)"><use data-c="29" xlink:href="#MJX-107-TEX-N-29"></use></g><g data-mml-node="mtext" transform="translate(5168,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(5418,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">在</text></g><g data-mml-node="mtext" transform="translate(6302,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mi" transform="translate(6552,0)"><use data-c="1D445" xlink:href="#MJX-107-TEX-I-1D445"></use></g><g data-mml-node="mtext" transform="translate(7311,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mtext" transform="translate(7561,0)"><text data-variant="normal" transform="scale(1,-1)" font-size="884px" font-family="serif">上</text></g><g data-mml-node="mtext" transform="translate(8445,0)"><text 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xlink:href="#MJX-107-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(8722.6,0)"><use data-c="221E" xlink:href="#MJX-107-TEX-N-221E"></use></g><g data-mml-node="mo" transform="translate(9722.6,0)"><use data-c="2C" xlink:href="#MJX-107-TEX-N-2C"></use></g><g data-mml-node="mtext" transform="translate(10167.2,0)"><use data-c="A0" xlink:href="#MJX-107-TEX-N-A0"></use></g><g data-mml-node="mfrac" transform="translate(10417.2,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="33" xlink:href="#MJX-107-TEX-N-33"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="34" xlink:href="#MJX-107-TEX-N-34"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(11357.2,0)"><use data-c="5D" xlink:href="#MJX-107-TEX-N-5D"></use></g></g></g></g></g></g></g></g></g></g></svg></mjx-container><script type="math/tex">\color{blue}\begin{align}&由\ 对任意的\ x_1,\ x_2\in R,\ 都有\ (x_1-x_2)[(f(x_1)-f(x_2)]<0\\&可知\ 函数\ f(x)在\ R\ 上为减函数.\\&\therefore当\ x<0\ 时,\ f(x)=-x+3-3a\ 为一次函数,\ 且为减函数,\ 则此时\ f(x)>f(0)=3-3a;\\&\quad\ 当\ x\geqslant0\ 时,\ 函数f(x)=-x^2+a\ 为二次函数,\ 也为减函数,\ 且有\ f（x)\leqslant f(0)=a\\&\quad\ 要使\ f(x)\ 在\ R\ 上为减函数,\ 则有\ a\leqslant 3-3a,\ 解得\ a\leqslant\frac34\\&\therefore a\ 的取值范围是\ (-\infty,\ \frac34]\end{align}</script></p>
<p>&nbsp;</p>
<h3 id='五函数的最值'>五、函数的最值</h3>
<h4 id='14-函数\n fx1xx⩾1−x22x1 \n的最大值为 2'>14. 函数<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="25.788ex" height="5.66ex" role="img" focusable="false" viewBox="0 -1500.8 11398.1 2501.6" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -2.264ex;"><defs><path id="MJX-108-TEX-N-A0" d=""></path><path id="MJX-108-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 -186T55 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<h4 id='16-求二次函数\n fxx2−2ax2 \n在\n\n 24 \n上的最小值'>16. 求二次函数<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="21.014ex" height="2.452ex" role="img" focusable="false" viewBox="0 -833.9 9288 1083.9" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-109-TEX-N-A0" d=""></path><path id="MJX-109-TEX-I-1D453" d="M118 -162Q120 -162 124 -164T135 -167T147 -168Q160 -168 171 -155T187 -126Q197 -99 221 27T267 267T289 382V385H242Q195 385 192 387Q188 390 188 397L195 425Q197 430 203 430T250 431Q298 431 298 432Q298 434 307 482T319 540Q356 705 465 705Q502 703 526 683T550 630Q550 594 529 578T487 561Q443 561 443 603Q443 622 454 636T478 657L487 662Q471 668 457 668Q445 668 434 658T419 630Q412 601 403 552T387 469T380 433Q380 431 435 431Q480 431 487 430T498 424Q499 420 496 407T491 391Q489 386 482 386T428 385H372L349 263Q301 15 282 -47Q255 -132 212 -173Q175 -205 139 -205Q107 -205 81 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xlink:href="#MJX-109-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(4714.3,0)"><use data-c="2212" xlink:href="#MJX-109-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(5714.6,0)"><use data-c="32" xlink:href="#MJX-109-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(6214.6,0)"><use data-c="1D44E" xlink:href="#MJX-109-TEX-I-1D44E"></use></g><g data-mml-node="mi" transform="translate(6743.6,0)"><use data-c="1D465" xlink:href="#MJX-109-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(7537.8,0)"><use data-c="2B" xlink:href="#MJX-109-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(8538,0)"><use data-c="32" xlink:href="#MJX-109-TEX-N-32"></use></g><g data-mml-node="mtext" transform="translate(9038,0)"><use data-c="A0" xlink:href="#MJX-109-TEX-N-A0"></use></g></g></g></svg></mjx-container><script type="math/tex">\ f(x)=x^2-2ax+2\ </script>在<mjx-container class="MathJax" jax="SVG" style="position: relative;"><svg xmlns="http://www.w3.org/2000/svg" width="6.652ex" height="2.262ex" role="img" focusable="false" viewBox="0 -750 2940 1000" xmlns:xlink="http://www.w3.org/1999/xlink" aria-hidden="true" style="vertical-align: -0.566ex;"><defs><path id="MJX-110-TEX-N-A0" d=""></path><path id="MJX-110-TEX-N-5B" d="M118 -250V750H255V710H158V-210H255V-250H118Z"></path><path id="MJX-110-TEX-N-32" d="M109 429Q82 429 66 447T50 491Q50 562 103 614T235 666Q326 666 387 610T449 465Q449 422 429 383T381 315T301 241Q265 210 201 149L142 93L218 92Q375 92 385 97Q392 99 409 186V189H449V186Q448 183 436 95T421 3V0H50V19V31Q50 38 56 46T86 81Q115 113 136 137Q145 147 170 174T204 211T233 244T261 278T284 308T305 340T320 369T333 401T340 431T343 464Q343 527 309 573T212 619Q179 619 154 602T119 569T109 550Q109 549 114 549Q132 549 151 535T170 489Q170 464 154 447T109 429Z"></path><path id="MJX-110-TEX-N-34" d="M462 0Q444 3 333 3Q217 3 199 0H190V46H221Q241 46 248 46T265 48T279 53T286 61Q287 63 287 115V165H28V211L179 442Q332 674 334 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&2\leqslant a\leqslant 4\\18-8a &a>4\end{cases}\end{align}</script></p>
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